We include three different types of boundary conditions: Dirichlet in \(d\mathcal {B}_D\) , Robin in \(d\mathcal {B}_C\)  and Neumann \(d\mathcal {B}_N\) . The Dirichlet boundary condition is to represent another body with constant, higher than ambient, temperature, which the heat sink \(\mathcal {B}\)  is attempting to dissipate. The Robin boundary condition represents a convective flux, which is proportional to the temperature \(u\) . The Neumann boundary condition is used to represent the symmetry of the domain. The geometry and location of the boundaries are shown in Figure REF . This example could be interpreted as a simplified model for the heat sink that is typically placed above a central processing unit (CPU) of a computer. To generate a ground truth solution for this problem we performed finite element simulations using the FEniCS library [1]} on a mesh with 1289 nodes and 2183 linear triangular elements. In this experiment we will attempt to solve the inverse problem of learning the temperature inside the domain \(\mathcal {B}\)  from the information of the temperature in the boundary \(d\mathcal {B}\) . A practical realization of this experiment could be achieved by using a thermal camera and attempt to predict the temperature inside the heat sink. We will again use the proposed \(\Delta \) -PINNs with and without physics, as well conventional PINNs to solve this problem. In all cases, we will not inform the type of boundary conditions applied, and instead we only provide temperature measurements \(u_i\)  at the boundary \(d\mathcal {B}\) . Thus, we will not consider the term \(MSE_{\rm boundary}\)  in the loss function defined in equation REF . In practice, this is important because the true form of the convective fluxes is often unknown or empirically approximated. For \(\Delta \) -PINNs we will approximate the Laplacian operator \(\Delta u\)  with triangular finite elements using the definition of discrete Laplacian \(\mbox{$A$}{}\)  shown in equation REF . Then, we can define the loss term:
\(MSE_{\rm PDE}(\mbox{$\theta $}{}) = \sum _i^R \left(\sum _{j \in N(i)} A_{ij} \hat{u}_j \right)^2\) 
